Two families of circulant nut graphs

Abstract

A circulant nut graph is a non-trivial simple graph whose adjacency matrix is a circulant matrix of nullity one such that its non-zero null space vectors have no zero elements. The study of circulant nut graphs was originally initiated by Basic et al. [Art Discrete Appl. Math. 5(2) (2021) #P2.01], where a conjecture was made regarding the existence of all the possible pairs (n, d) for which there exists a d-regular circulant nut graph of order n. Later on, it was proved by Damnjanovic and Stevanovic [Linear Algebra Appl. 633 (2022) 127-151] that for each odd t ? 3 such that t ?/10 1 and t ?/18 15, the 4t-regular circulant graph of order n with the generator set {1, 2, 3,..., 2t+1}\{t}) must necessarily be a nut graph for each even n ? 4t + 4. In this paper, we extend these results by constructing two families of circulant nut graphs. The first family comprises the 4t-regular circulant graphs of order n which correspond to the generator sets {1, 2,..., t?1} ? {n/4, n/4 + 1} ? {n/2?(t?1),..., n/2 ? 2, n/2 ? 1}, for each odd t ? N and n ? 4t + 4 divisible by four. The second family consists of the 4t-regular circulant graphs of order n which correspond to the generator sets {1, 2,..., t?1} ? {n+2/4, n+6/4} ? {n/2 ?(t?1),..., n/2?2, n/2?1}, for each t ? N and n ? 4t + 6 such that n ?4 2. We prove that all of the graphs which belong to these families are indeed nut graphs, thereby fully resolving the 4t-regular circulant nut graph order-degree existence problem whenever t is odd and partially solving this problem for even values of t as well.